What if the volume of a cylinder as a function of it's height/radius? Full question in the description box below.

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What if the volume of a cylinder as a function of it's height/radius? Full question in the description box below.

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Answer 1 · 2018-01-04T19:36:00

$V_{1} (h) = \frac{1}{16} h (144 - h^{2}) {cm}^{3}$ $V_1(h) = 1/16 h(144-h^2)" cm"^3" "$ for $h \in [0, 12]$ $h in [0, 12]$

$V_{2} (r) = 2 π r^{2} \sqrt{36 - r^{2}} {cm}^{3}$ $V_2(r) = 2pir^2 sqrt(36-r^2)" cm"^3" "$ for $r \in [0, 6]$ $r in [0, 6]$

Explanation:

If we take a vertical slice through the centre of the sphere and cylinder, then it looks like this:

enter image source here

Where $r$ $r$ is the radius of the cylinder, $R$ $R$ the radius of the sphere and $h$ $h$ the height of the cylinder.

Then the volume of the cylinder is the area of its base, multiplied by the height. That is:

$V = h π r^{2}$ $V = h pi r^2$

Note that the radius $r$ $r$ varies as the height $h$ $h$ varies, always satisfying Pythagoras formula:

${(2 R)}^{2} = h^{2} + {(2 r)}^{2}$ $(2R)^2 = h^2+(2r)^2$

That is:

$4 R^{2} = h^{2} + 4 r^{2}$ $4R^2 = h^2+4r^2$

Hence we can write $r$ $r$ in terms of $h$ $h$ or $h$ $h$ in terms of $r$ $r$ :

$r = \frac{1}{4} \sqrt{4 R^{2} - h^{2}}$ $r = 1/4 sqrt(4R^2-h^2)$

$h = \sqrt{4 R^{2} - 4 r^{2}} = 2 \sqrt{R^{2} - r^{2}}$ $h = sqrt(4R^2-4r^2) = 2sqrt(R^2-r^2)$

We can substitute these formulae into our prior formula for the volume of the cylinder to find:

$V_{1} (h) = h π r^{2} = h π {(\frac{1}{4} \sqrt{4 R^{2} - h^{2}})}^{2} = \frac{1}{16} h (4 R^{2} - h^{2})$ $V_1(h) = h pi r^2 = h pi (1/4 sqrt(4R^2-h^2))^2 = 1/16 h(4R^2-h^2)$

$V_{2} (r) = h π r^{2} = (2 \sqrt{R^{2} - r^{2}}) π r^{2} = 2 π r^{2} \sqrt{R^{2} - r^{2}}$ $V_2(r) = h pi r^2 = (2sqrt(R^2-r^2)) pi r^2 = 2pir^2 sqrt(R^2-r^2)$

Finally, substituting $R = 6$ $R=6$ (cm) we get:

$V_{1} (h) = \frac{1}{16} h (144 - h^{2}) {cm}^{3}$ $V_1(h) = 1/16 h(144-h^2)" cm"^3" "$ for $h \in [0, 12]$ $h in [0, 12]$

$V_{2} (r) = 2 π r^{2} \sqrt{36 - r^{2}} {cm}^{3}$ $V_2(r) = 2pir^2 sqrt(36-r^2)" cm"^3" "$ for $r \in [0, 6]$ $r in [0, 6]$

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What if the volume of a cylinder as a function of it's height/radius? Full question in the description box below.

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